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We get stationary solutions of a free stochastic partial differential equation. As an application, we prove equality of non-microstate and microstate free entropy dimensions under a Lipschitz like condition on conjugate variables, assuming also R^\omega\ embeddability. This includes an N-tuple of q-Gaussian random variables e.g. for |q|N\leq 0.13. Source: http://arxiv.org/abs/1008.4742v3

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The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is based on factorization of a non-homogeneous first order differential operator to products consisting of a scalar function, a homogeneous first order differential operator and the reciprocal of the scalar function. The factorization procedure is utilized to show that all totally linear differential equations of first order can be transformed to each... Source: http://arxiv.org/abs/math/9906085v1

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We determine a considerable class of nonlinear partial differential equation systems which have global regular solutions. Uniqueness is not a direct general consequence of this method. The scheme can be applied to the incompressible Navier Stokes equation. Topics: Mathematics, Analysis of PDEs Source: http://arxiv.org/abs/1501.05849

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For the linear partial differential equation $P(\partial_x,\partial_t)u=f(x,t)$, where $x\in\mathbb{R}^n,\;t\in\mathbb{R}^1$, with $P(\partial_x,\partial_t)$ is $\prod^m_{i=1}(\frac{\partial}{\partial{t}}-a_iP(\partial_x))$ or $\prod^m_{i=1}(\frac{\partial^2}{\partial{t^2}}-a_i^2P(\partial_x))$, the authors give the analytic solution of the cauchy problem using the abstract operators $e^{tP(\partial_x)}$ and $\frac{\sinh(tP(\partial_x)^{1/2})}{P(\partial_x)^{1/2}}$. By representing the... Source: http://arxiv.org/abs/1010.0761v2

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In this paper we investigate a nonlinear stochastic partial differential equation (spde in short) perturbed by a space-correlated Gaussian noise in arbitrary dimension $d\geq1$, with a non-Lipschitz coefficient noisy term. The equation studied coincides in one dimension with the stochastic Burgers equation. Existence of a weak solution is established through an approximation procedure. Source: http://arxiv.org/abs/1104.5416v1

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In this paper, we prove existence, uniqueness and regularity for a class of stochastic partial differential equations with a fractional Laplacian driven by a space-time white noise in dimension one. The equation we consider may also include a reaction term. Source: http://arxiv.org/abs/math/0510107v1

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"Note on a Partial Differential Equation of the First Order§" is an article from The Annals of Mathematics, Volume 4 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967127 Source: http://www.jstor.org/stable/10.2307/1967127

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In this article we consider cloaking for a quasi-linear elliptic partial differential equation of divergence type defined on a bounded domain in $\mathbb{R}^N$ for $N=2,3$. We show that a perfect cloak can be obtained via a singular change of variables scheme and an approximate cloak can be achieved via a regular change of variables scheme. These approximate cloaks though non-degenerate are anisotropic. We also show, within the framework of homogenization, that it is possible to get isotropic... Topics: Analysis of PDEs, Mathematics Source: http://arxiv.org/abs/1704.02714

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The elliptic 2-Hessian equation is a fully nonlinear partial differential equation (PDE) that is related to intrinsic curvature for three dimensional manifolds. We introduce two numerical methods for this PDE: the first is provably convergent to the viscosity solution, and the second is more accurate, and convergent in practice but lacks a proof. The PDE is elliptic on a restricted set of functions: a convexity type constraint is needed for the ellipticity of the PDE operator. Solutions with... Topics: Mathematics, Numerical Analysis Source: http://arxiv.org/abs/1502.04969

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We consider the generalized parabolic Anderson equation (gPAM) in 2 dimensions with periodic boundary. This is an example of a singular semilinear stochastic partial differential equations, solutions of which require renormalization and have only be understood recently via Hairer's regularity structures and, in some cases equivalently, paracontrollled distributions due to Gubinelli, Imkeller and Perkowski. In the present paper we describe the law of gPAM, by establishing a Stroock{Varadhan type... Topics: Probability, Mathematics Source: http://arxiv.org/abs/1409.4250

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In this article we introduce a Partial Differential Equation (PDE) for the rank one convex envelope. Rank one convex envelopes arise in non-convex vector valued variational problems \cite{BallElasticity, kohn1986optimal1, BallJames87, chipot1988equilibrium}. More generally, we study a PDE for directional convex envelopes, which includes the usual convex envelope \cite{ObermanConvexEnvelope} and the rank one convex envelope as special cases. Existence and uniqueness of viscosity solutions to the... Topics: Analysis of PDEs, Mathematics Source: http://arxiv.org/abs/1605.03155

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A nonlinear parabolic differential equation with a quadratic nonlinearity is presented which has at least one equilibrium. The linearization about this equilibrium is asymptotically stable, but by using a technique inspired by H. Fujita, we show that the equilibrium is unstable in the nonlinear setting. The perturbations used have the property that they are small in every $L^p$ norm, yet they result in solutions which fail to be global. Source: http://arxiv.org/abs/0704.3989v2

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We explore Ito stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic... Source: http://arxiv.org/abs/math/0509166v1

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In a series of papers Barron, Goebel, and Jensen studied Partial Differential Equations (PDE)s for quasiconvex (QC) functions \cite{barron2012functions, barron2012quasiconvex,barron2013quasiconvex,barron2013uniqueness}. To overcome the lack of uniqueness for the QC PDE, they introduced a regularization: a PDE for $\e$-robust QC functions, which is well-posed. Building on this work, we introduce a stronger regularization which is amenable to numerical approximation. We build convergent finite... Topics: Analysis of PDEs, Numerical Analysis, Mathematics Source: http://arxiv.org/abs/1612.06813

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In this course, we will mainly consider the case of free particles, in which V = 0 (i.e., the homogeneous Schr�odinger equation). In the case of free particles, there is an important family of solutions to (1.0.1), namely the free waves. The free wave solutions provide some important intuition about how solutions to the homogeneous Schr�odinger equation behave. Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics Source: http://www.flooved.com/reader/1607

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The aim of this paper is to provide an overview of recent development related to Bregman distances outside its native areas of optimization and statistics. We discuss approaches in inverse problems and image processing based on Bregman distances, which have evolved to a standard tool in these fields in the last decade. Moreover, we discuss related issues in the analysis and numerical analysis of nonlinear partial differential equations with a variational structure. For such problems Bregman... Topics: Optimization and Control, Mathematics Source: http://arxiv.org/abs/1505.05191

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In this paper, we introduce the concept of Developmental Partial Differential Equation (DPDE), which consists of a Partial Differential Equation (PDE) on a time-varying manifold with complete coupling between the PDE and the manifold's evolution. In other words, the manifold's evolution depends on the solution to the PDE, and vice versa the differential operator of the PDE depends on the manifold's geometry. DPDE is used to study a diffusion equation with source on a growing surface whose... Topics: Optimization and Control, Mathematics Source: http://arxiv.org/abs/1508.04648

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In this review paper, we explain how to apply Renormalization Group ideas to the analysis of the long-time asymptotics of solutions of partial differential equations. We illustrate the method on several examples of nonlinear parabolic equations. We discuss many applications, including the stability of profiles and fronts in the Ginzburg-Landau equation, anomalous scaling laws in reaction-diffusion equations, and the shape of a solution near a blow-up point. Source: http://arxiv.org/abs/chao-dyn/9411015v1

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We present a new topological method for the study of the dynamics of dissipative PDE's. The method is based on the concept of the self-consistent apriori bounds, which allows to justify rigorously the Galerkin projection. As a result we obtain a low-dimensional system of ODE's subject to rigorously controlled small perturbation from the neglected modes. To this ODE's we apply the Conley index to obtain information about the dynamics of the PDE under consideration. As an application we present a... Source: http://arxiv.org/abs/math/0005247v1

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This book has developed from courses of lectures given by the author over a period of years to the students of the Moscow PhysicoTechnical Institute. It is intended for the students having basic knowledge of mathematical analysis, algebra and the theory of ordinary differential equations to the extent of a university course. Except Chapter I, where some general questions regarding partial differential equations have been examined, the material has been arranged so as to correspond to the basic...favoritefavoritefavoritefavoritefavorite ( 1 reviews ) Topics: mathematics, partial differential equations, elliptic equations, hyperbolic equations, parabolic...

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We propose a new algebraic approach to study compatibility of partial differential equations. The approach uses concepts from commutative algebra, algebraic geometry and Gr\"obner bases to clarify crucial notions concerning compatibility such as passivity (involution) and reducibility. One obtains sufficient conditions for a differential system to be passive and prove that such systems generate manifolds in the jet space. Some examples of constructions of passive systems associated with... Topics: Dynamical Systems, Mathematical Physics, Mathematics Source: http://arxiv.org/abs/1611.05441

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This book encompasses both traditional and modern methods treating partial differential equation (PDE) of first order and second order. There is a balance in making a selfcontained mathematical text and introducing new subjects. The Lie algebras of vector fields and their algebraic-geometric representations are involved in solving overdetermined of PDE and getting integral representation of stochastic differential equations (SDE). It is addressing to all scientists using PDE in treating... Source: http://arxiv.org/abs/1004.2134v1

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If the subject of ordinary di_erential equations is large, this is enormous. I am going to examine only one corner of it, and will develop only one tool to handle it: Separation of Variables. Another major tool is the method of characteristics and I�ll not go beyond mentioning the word. When I develop a technique to handle the heat equation or the potential equation, don�t think that it stops there. The same set of tools will work on the Schroedinger equation in quantum mechanics and on the... Topics: Physics, Condensed Matter, Mathematical Methods in Physics, Structural, Mechanical, and Thermal... Source: http://www.flooved.com/reader/3023

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The solution to the nonlinear output regulation problem requires one to solve a first order PDE, known as the Francis-Byrnes-Isidori (FBI) equations. In this paper we propose a method to compute approximate solutions to the FBI equations when the zero dynamics of the plant are hyperbolic and the exosystem is two-dimensional. With our method we are able to produce approximations that converge uniformly to the true solution. Our method relies on the periodic nature of two-dimensional analytic... Source: http://arxiv.org/abs/1106.5815v1

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These are introductory notes on ordinary and partial differential equations. Assumed background is calculus and a little physics. Linear algebra is not assumed, and is introduced here in four of the lectures. Those four lectures have been used in the Engineering Mathematics course at Cornell University for several years. Lecture Notes Collection FreeScience.info ID911 Obtained from http://www.math.cornell.edu/~bterrell/dn.pdf http://www.freescience.info/go.php?pagename=books&id=911 Topics: Differential Equations, "

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This is a survey on Chaos in Partial Differential Equations. First we classify soliton equations into three categories: 1. (1+1)-dimensional soliton equations, 2. soliton lattices, 3. (1+n)-dimensional soliton equations (n greater than 1). A systematic program has been established by the author and collaborators, for proving the existence of chaos in soliton equations under perturbations. For each category, we pick a representative to present the results. Then we review some initial results on... Source: http://arxiv.org/abs/math/0205114v1

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The asymptotic theory of the reduced wave equation and Maxwell's equations for high frequencies is presented. The theory is applied to representative problems involving reflection, transmission, and diffraction in homogeneous and inhomogeneous media. The report contains few new results. It is intended to unify and summarize the existing literature on the subject. Topics: DTIC Archive, Lewis, Robert M, NEW YORK UNIV NY COURANT INST OF MATHEMATICAL SCIENCES, *PARTIAL...

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We describe hypergeometric solutions of the quantum differential equation of the cotangent bundle of a gl_n partial flag variety. These hypergeometric solutions manifest the Landau-Ginzburg mirror symmetry for the cotangent bundle of a partial flag variety. Source: http://arxiv.org/abs/1301.2705v1

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Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by the author in recent years, with emphasis on physical equations such as: the Calogero-Sutherland model of quantum many-body system in one-dimension, the Maxwell equations, the free Dirac equations, the generalized acoustic system, the Kortweg and de Vries... Source: http://arxiv.org/abs/1205.6535v1

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In this paper, we investigate a nonlinear partial differential equation, arising from a model of cellular proliferation. This model describes the production of blood cells in the bone marrow. It is represented by a partial differential equation with a retardation of the maturation variable and a distributed temporal delay. Our aim is to prove that the behaviour of primitive cells influences the global behaviour of the population. Source: http://arxiv.org/abs/0904.2472v1

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This paper develops one of the methods for study of nonlinear Partial Differential equations. We generalize Sato equation and represent the algorithm for construction of some classes of nonlinear Partial Differential Equations (PDE) together with solutions parameterized by the set of arbitrary functions. Source: http://arxiv.org/abs/nlin/0202053v3

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We show that for any semilinear partial differential equation of order m, the infinitesimals of the independent variables depend only on the independent variables and, if m>1 and the equation is also linear in its derivatives of order m-1 of the dependent variable, then the infinitesimal of the dependent variable is at most linear on the dependent variable. Many examples of important partial differential equations in Analysis, Geometry and Mathematical - Physics are given in order to... Source: http://arxiv.org/abs/0803.0865v2

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In this work we propose a mechanism for converting the spectral problem of vertex models transfer matrices into the solution of certain linear partial differential equations. This mechanism is illustrated for the $U_q[\widehat{\mathfrak{sl}}(2)]$ invariant six-vertex model and the resulting partial differential equation is studied for particular values of the lattice length. Topics: Mathematics, Nonlinear Sciences, Analysis of PDEs, Mathematical Physics, Exactly Solvable and... Source: http://arxiv.org/abs/1403.0425

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This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. We first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then we establish necessary and sufficient optimality conditions of the... Topics: Optimization and Control, Mathematics Source: http://arxiv.org/abs/1610.02486

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A general principle for error estimation is described which can be applied to different types of partial differential equations. Particular attention is paid to nonlinear problems. With a programmed procedure based on this estimation principle, error bounds are calculated for boundary value problems involving the differential equation -deltau + f(x,y,u) = 0. Topics: DTIC Archive, BOEING SCIENTIFIC RESEARCH LABS SEATTLE WA, *DIFFERENTIAL EQUATIONS, *INEQUALITIES,...

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The authors first solve the equation dX + aXdt = dN, where dN represents a Poisson process, and then generalize to a Levy process. Finally, they solve a linear partial differential equation DX = dL in strong distribution, meaning that the second member dL is a distribution process, generalization of Levy process on R. The results are then applied to wave propagation in underwater acoustics, and spatial correction is determined. (Author) Topics: DTIC Archive, DEBrucq,D, NORTH CAROLINA UNIV AT CHAPEL HILL DEPT OF STATISTICS, *Linear...

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In this paper we consider an approximation scheme for an optimal control problem described by a hyperbolic partial-functional differential equation used to model the elastic motion of a viscoelastic body of Boltzmann type. The method is based on combined finite element/averaging approximations. We present theoretical and numerical results for a problem with quadratic cost functional. Topics: DTIC Archive, Burns, J A, BROWN UNIV PROVIDENCE RI LEFSCHETZ CENTER FOR DYNAMICAL SYSTEMS,...

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An important application of the higher partial derivatives is that they are used in partial di_erential equations to express some laws of physics which are basic to most science and engineering subjects. In this section, we will give examples of a few such equations. The reason is partly cultural, so you meet these equations early and learn to recognize them, and partly technical: to give you a little more practice with the chain rule and computing higher derivatives. Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Partial Differential Equations (PDEs),... Source: http://www.flooved.com/reader/1811

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In this work, we investigate a unique solvability of a direct and inverse source problem for a time-fractional partial differential equation with the Caputo and Bessel operators. Using spectral expansion method, we give explicit forms of solutions to formulated problems in terms of multinomial Mittag-Leffler and first kind Bessel functions. Topics: Analysis of PDEs, Mathematics Source: http://arxiv.org/abs/1611.01624

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We study a higher order parabolic partial differential equation that arises in the context of condensed matter physics. It is a fourth order semilinear equation whose nonlinearity is the determinant of the Hessian matrix of the solution. We consider this model in a bounded domain of the real plane and study its stationary solutions both when the geometry of this domain is arbitrary and when it is the unit ball and the solution is radially symmetric. We also consider the initial-boundary value... Topics: Analysis of PDEs, Mathematics, Classical Analysis and ODEs Source: http://arxiv.org/abs/1503.06732

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We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation as, e.g., the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical "one-dimensional" results we discuss new features of the considered equation like, e.g., an analogue of the Cauchy integral... Source: http://arxiv.org/abs/0706.1744v2